Abstract

Modern inventory management, environmental concerns, and supply chain management challenges have made the two-warehouse system a necessary and strategic approach. Early studies explored demand patterns, this system increases flexibility, reduces risk, and improves efficiency, while also facilitating the implementation of modern strategies such as green sustainability practices, the use of renewable resources, preservation technologies, and carbon-efficient supply chain management. Studies began with exploring demand patterns, backlogging, LIFO and FIFO dispatching systems, trade credit financing, and various modelling methods related to product deterioration over time. More recent research has incorporated factors aimed at improving ecological performance, such as reverse logistics, carbon emission control, sustainable transportation, preservation methods, and green technologies. This review paper emphasizes the urgent need to optimize sustainable two-warehouse inventory models within an economically beneficial, sustainable, smart, and climate-resilient framework, and to integrate preservation and green technologies, carbon emission regulations, and multi-echelon supply chain frameworks into future research and practice. 

Author

Preeti Mehta1, Akshya Yadav2*, A. K. Malik3  ( Pages 67-96 )
Email:drpreeti@bnuniversity.ac.in
Affiliation:Department of Mathematics and Statistics, B.N. University, Udaipur, Rajasthan, India      DOI: https://doi.org/10.58517/IJSE.2025.04201

Keyword

Inventory, Demand, Deterioration, Two warehouses

References

1.      warranty and green level of the product with nonlinear demand via optimal control theory and Artificial Hummingbird Algorithm. Scientific Reports, 14(1), 10809.

2.      Das, S. C. (2024). Low Financing Trade Credit Inventory for High Deterioration under Time Dependent Holding Cost. Indian Journal of Science and Technology, 17(30), 3093-3099.

3.      Yadav, R., Sharma, A. K., & Singh, P. T. (2024). A Production inventory model for fractionally time-dependent demand rate with weibull deterioration and partially backlogged items. Asian Research Journal of Mathematics, 20(8), 102-118.

4.      Kumar, P., Saxena, A., & Kumar, K. (2024, December). Inventory Model for Two-Warehouse System Featuring Exponential Time-Varying Demand. In 2024 13th International Conference on System Modeling & Advancement in Research Trends (SMART) (pp. 508-512). IEEE.

5.      Mahalakshmi, G., & Chitra, D. (2024). Two Warehouse Eoq Model For Imperfect Quality And Reworkable Items With Parabolic Time Dependent Demand & Non-Linear Holding Cost Under Trade Credit Financing Policy Subject To Partial Backlogging. Educational Administration: Theory and Practice, 30(3), 2450-2462.

6.      Kaur, P., Chand, M., Kaur, D., & Rakshit, M. (2024). Inventory Modeling for Deteriorating Items with Stock Dependent Demand, Shortages and Inflation Under Two-Warehouse Storage Management System in a Fuzzy Environment. Complexity Analysis and Applications, 1(1), 14-24.

7.      Ahmed, A., Kummari, K., Bandaru, R., & Shukla, R. (2024). Two-warehouse inventory model: interval valued costs, advanced payment, stock-dependent demand, and partial backlogging. European Journal of Pure and Applied Mathematics, 17(3), 2028-2054.

8.      Zhong, Z., Yuan, M., & He, Z. (2024). Data-Driven Algorithms for Two-Location Inventory Systems. Systems, 12(5), 153.

9.      Tajik, M., Hajimolana, S. M., & Daneshvar Kakhki, M. (2024). Production Planning Optimization in a Two-Echelon Multi-Product Supply Chain with Discrete Delivery and Storage at Manufacturer’s Warehouse. Mathematics, 12(13), 1986.

10.   Tan, Y., Gu, L., Xu, S., & Li, M. (2024). Supply Chain Inventory Management from the Perspective of “Cloud Supply Chain”—A Data Driven Approach. Mathematics, 12(4), 573.

11.   Attia, E. A., Miah, M. M., Arif, A. S., AlArjani, A., Hasan, M., & Uddin, M. S. (2024). A Novel Model for Economic Recycle Quantity with Two-Level Piecewise Constant Demand and Shortages. Computation, 12(1), 13.

12.   Kumar, G., Bera, S., Samanta, G., & Maiti, M. (2024). Optimal profit in two-level trade credit EOQ model with default risk and reminder cost under finite time horizon having time-dependent demand and deterioration. RAIRO-Operations Research, 58(5), 3895-3921.

13.   Keswani, M., & Khedlekar, U. (2024). Optimizing pricing and promotions for sustained profitability in declining markets: A Green-Centric inventory model. Data Science in Finance and Economics, 4(1), 83-131.

14.   Kumar, A., Yadav, A. S., & Yadav, D. (2024). A two-storage production inventory model for deteriorating items with time and selling price dependent demand using flower pollination optimization. Journal of Research Administration Society of Research Administrator’s International, 6(1), 2950-2966.

15.   Limi, A., Rangarajan, K., Rajadurai, P., Akilbasha, A., & Parameswari, K. (2024). Three warehouse inventory model for non-instantaneous deteriorating items with quadratic demand, time-varying holding costs and backlogging over finite time horizon. Ain Shams Engineering Journal, 15(7), 102826.

16.   Pal, D., Manna, A. K., Ali, I., Roy, P., & Shaikh, A. A. (2024). A two-warehouse inventory model with credit policy and inflation effect. Decision Analytics Journal, 10, 100406.

17.   Das, D., & Samanta, G. C. (2024). An eoq model for two warehouse system during lock-down considering linear time dependent demand. Transactions on Quantitative Finance and Beyond, 1(1), 15-28.

18.   Pathak, K., Yadav, A. S., & Agarwal, P. (2024). Enhancing two-warehouse inventory models for perishable goods: time-price dependent demand, inflation, and partial backlogging. IAENG International Journal of Applied Mathematics, 54(6), 1089-1101.

19.   Yadav, K. K., Yadav, A. S., & Bansal, S. (2024). Optimization of an inventory model for deteriorating items assuming deterioration during carrying with two-warehouse facility. Reliability: Theory & Applications, 19(3 (79)), 442-459.

20.   Inaniyan, R., & Kumar, G. (2024). A comprehensive analysis of linear and non-linear deterioration rates in dual-warehouse inventory systems. Decision Analytics Journal, 10, 100396.

21.   Parida, S., Acharya, M., & Bohidar, C. (2024). Two-Warehouse Economic Order Quantity Model with Controllable Greenhouse Gas Emissions. Procedia Computer Science, 235, 2342-2352.

22.   Alamri, A. A. (2023). A sustainable closed-loop supply chains inventory model considering optimal number of remanufacturing times. Sustainability, 15(12), 9517.

23.   Momena, A. F., Haque, R., Rahaman, M., & Mondal, S. P. (2023). A two-storage inventory model with trade credit policy and time-varying holding cost under quantity discounts. Logistics, 7(4), 77.

24.   Safiya, M., Gopinath, D., & Lingam, G. S. (2023). Inventory control problem using metaheuristic techniques: application to two warehouse inventory model. Mathematical Statistician and Engineering Applications, 72(2), 91-105.

25.   Sebatjane, M., & Adetunji, O. (2024). A four-echelon supply chain inventory model for growing items with imperfect quality and errors in quality inspection. Annals of operations research, 335(1), 327-359.

26.   Amutha, R. (Year). An Inventory Model for Deteriorating Items with Two Parameter Weibull Demand under Inflation. International Journal of Science and Research, 12(6) 342-346.

27.   Palanivel, M., & Vetriselvi, S. (2024). Optimization of a Two-Warehouse EOQ Model for Non-Instantaneous Deteriorating Items with Polynomial Demand, Advance Payment, and Shortages. Contemporary Mathematics, 2770-2781.

28.   Mayan, M. K., Martin, N., Miriam, M. R., & Jayaraman, S. (2023). Inventory model with climate change impacts and green sustainability cost parameters. In E3S Web of Conferences (Vol. 405, p. 04009). EDP Sciences.

29.   Motla, R., Kumar, A., Saxena, N., & Sana, S. S. (2023). Inventory optimization in a green environment with two warehouses. Innovation and Green Development, 2(4), 100087.

30.   Jagadeesan, V., Rajamanickam, T., Schindlerova, V., Subbarayan, S., & Cep, R. (2023). A study on two-warehouse inventory systems with integrated multi-purpose production unit and partitioned rental warehouse. Mathematics, 11(18), 3986.

31.   Nurhasril, N. B. (2021). A two-warehouse inventory model with rework process and time-varying demand (Master's thesis, University of Malaya (Malaysia)).

32.   Shah, N. H., Keswani, M., Khedlekar, U. K., & Prajapati, N. M. (2024). Non-instantaneous controlled deteriorating inventory model for stock-price-advertisement dependent probabilistic demand under trade credit financing. Opsearch, 61(1), 421-459.

33.   Padiyar, S. V. S., Gupta, V., & Rajput, N. (2023). Multi-echelon supply chain inventory model for perishable items with fuzzy deterioration rate and imperfect production with two-warehouse under inflationary environment. International Journal of Business Performance and Supply Chain Modelling, 14(2), 144-172.

34.   Choudri, V., & Sivashankari, C. K. (2023). PRODUCTION INVENTORY MODELS WITH INTEGRATED STOCK AND PRICE DEPENDENT DEMANDS FOR DETERIORATIVE ITEMS. Pesquisa Operacional, 43, e265586.

35.   Singhal, S., Kumar, V., & Raghav, J. S. (2022). A two warehouse inventory model for deteriorating items with multivariate demand and backlogging. GANITA, 72(1), 39-47.

36.   Patil, K., Jain, A., & Yadav, R. (2022).An inventory model with partial backordering, Weibull distribution deterioration under two level of storage. NeuroQuantology, 20(13), 4125–4133.

37.   Kaushik, J. I. T. E. N. D. R. A. (2022). Development of inventory models for deteriorating items considering uniform, price and time-dependent demand-a review. Adv. Appl. Math. Sci, 21(7), 4083-4096.

38.   Berretta, R., & Paam, P. Bi-Objective Model for Multi-Warehouse Inventory for Fresh Produce with Transhipment.

39.   Kumar, N., Dahiya, S., & Kumar, S. (2022). Two warehouse inventory model for deteriorating items with fixed shelf-life stock-dependent demand and partial backlogging. J. Math. Comput. Sci., 12, Article-ID.

40.   Sebatjane, M. (2022). The impact of preservation technology investments on lot-sizing and shipment strategies in a three-echelon food supply chain involving growing and deteriorating items. Operations Research Perspectives, 9, 100241.

41.   Khan, M. A. A., Halim, M. A., AlArjani, A., Shaikh, A. A., & Uddin, M. S. (2022). Inventory management with hybrid cash-advance payment for time-dependent demand, time-varying holding cost and non-instantaneous deterioration under backordering and non-terminating situations. Alexandria Engineering Journal, 61(11), 8469-8486.

42.   Priyan, S., Mala, P., & Palanivel, M. (2022). A cleaner EPQ inventory model involving synchronous and asynchronous rework process with green technology investment. Cleaner Logistics and Supply Chain, 4, 100056.

43.   Rajeswari, S., Sugapriya, C., & Nagarajan, D. (2022). An analysis of uncertain situation and advance payment system on a double-storage fuzzy inventory model. Opsearch, 59(1), 20-40.

44.   Mashud, A. H. M., Pervin, M., Mishra, U., Daryanto, Y., Tseng, M. L., & Lim, M. K. (2021). A sustainable inventory model with controllable carbon emissions in green-warehouse farms. Journal of Cleaner Production, 298, 126777.

45.   Malik, A. K., & Garg, H. (2021). An improved fuzzy inventory model under two warehouses. Journal of Artificial Intelligence and Systems, 3(1), 115-129.

46.   Duary, A., Das, S., Arif, M. G., Abualnaja, K. M., Khan, M. A. A., Zakarya, M., & Shaikh, A. A. (2022). Advance and delay in payments with the price-discount inventory model for deteriorating items under capacity constraint and partially backlogged shortages. Alexandria Engineering Journal, 61(2), 1735-1745.

47.   Jani, M. Y., Betheja, M. R., Chaudhari, U., & Sarkar, B. (2021). Optimal investment in preservation technology for variable demand under trade-credit and shortages. Mathematics, 9(11), 1301.

48.   Nath, B. K., & Sen, N. (2021). A completely backlogged two-warehouse inventory model for non-instantaneous deteriorating items with time and selling price dependent demand. International Journal of Applied and Computational Mathematics, 7(4), 145.

49.   Garg, S. K., Kumar, V., & Sisodia, N. K. (2021). Two-warehouse inventory model for perishable items with variable demand under inflationary environment. In Recent Trends in Industrial and Production Engineering: Select Proceedings of ICCEMME 2021 (pp. 57-68). Singapore: Springer Singapore.

50.   Saxena, P., Singh, C., & Sharma, K. (2020). Green design and product stewardship approach for two-warehouse inventory model. Indian J Sci Technol, 13(37), 3850-3870.

51.   Kumar, M., Chauhan, A., Singh, S. J., & Sahni, M. (2020). An inventory model on preservation technology with trade credits under demand rate dependent on advertisement, time and selling price. Universal Journal of Accounting and Finance, 8(3), 65-74.

52.   Hishamuddin, I., Supadi, S. S., & Omar, M. (2020). Improved preservation technology for non-instantaneous deteriorating inventory using boundary condition estimation. Applied and Computational Mathematics, 9(4), 118-129.

53.   Chakraborty, D., Jana, D. K., & Roy, T. K. (2020). Multi-warehouse partial backlogging inventory system with inflation for non-instantaneous deteriorating multi-item under imprecise environment. Soft Computing, 24, 14471-14490.

54.   Lopes, C., Correia, A., Costa e Silva, E., Monteiro, M., & Borges Lopes, R. (2020). Inventory models with reverse logistics for assets acquisition in a liquefied petroleum gas company. Journal of Mathematics in Industry, 10, 1-15.

55.   Yadav, A. S., & Swami, A. (2019). An inventory model for non-instantaneous deteriorating items with variable holding cost under two-storage. International journal of procurement management, 12(6), 690-710.

56.   Tayal, S., Singh, S. R., & Attri, A. K. (2019). Two levels of storage model for deteriorating items, stock dependent demand and partial backlogging with both rented warehouses. International Journal of Process Management and Benchmarking, 9(4), 485-498.

57.   Indrajitsingha, S. K., Samanta, P. N., & Misra, U. K. (2019). A fuzzy two-warehouse inventory model for single deteriorating item with selling-price-dependent demand and shortage under partial-backlogged condition. Applications and Applied Mathematics: An International Journal (AAM), 14(1), 36.

58.   Rahimi-Ghahroodi, S., Al Hanbali, A., Vliegen, I. M. H., & Cohen, M. A. (2019). Joint optimization of spare parts inventory and service engineers staffing with full backlogging. International journal of production economics, 212, 39-50.

59.   Canyakmaz, C., Özekici, S., & Karaesmen, F. (2019). An inventory model where customer demand is dependent on a stochastic price process. International Journal of Production Economics, 212, 139-152.

60.   Kumar, B. A., Paikray, S. K., & Dutta, H. (2020). Cost optimization model for items having fuzzy demand and deterioration with two-warehouse facility under the trade credit financing. AIMS Math, 5(2), 1603-1620.

61.   Cárdenas-Barrón, L. E., Shaikh, A. A., Tiwari, S., & Treviño-Garza, G. (2020). An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit. Computers & Industrial Engineering, 139, 105557.

62.   Kumar, A., & Chanda, U. (2018). Two-warehouse inventory model for deteriorating items with demand influenced by innovation criterion in growing technology market. Journal of Management Analytics, 5(3), 198-212.

63.   Panda, G. C., Khan, M. A. A., & Shaikh, A. A. (2019). A credit policy approach in a two-warehouse inventory model for deteriorating items with price-and stock-dependent demand under partial backlogging. Journal of Industrial Engineering International, 15(1), 147-170.

64.   Tyagi, B., Yadav, A. S., Sharma, S., & Swami, A. Two ware-house Fuzzy Inventory model for Deteriorating Items with Ramp Type Demand and Shortages.

65.   Khan, M. A. A., Shaikh, A. A., Panda, G. C., & Konstantaras, I. (2019). Two-warehouse inventory model for deteriorating items with partial backlogging and advance payment scheme. RAIRO-operations Research, 53(5), 1691-1708.

66.   Alamri, A. A., & Syntetos, A. A. (2018). Beyond LIFO and FIFO: Exploring an allocation-in-fraction-out (AIFO) policy in a two-warehouse inventory model. International Journal of Production Economics, 206, 33-45.

67.   Mashud, A., Khan, M., Uddin, M., & Islam, M. (2018). A non-instantaneous inventory model having different deterioration rates with stock and price dependent demand under partially backlogged shortages. Uncertain Supply Chain Management, 6(1), 49-64.

68.   Malik, A. K., Vedi, P., & Kumar, S. (2018). An inventory model with time varying demand for non-instantaneous deteriorating items with maximum life time. International Journal of Applied Engineering Research, 13(9), 7162-7167.

69.   Tiwari, S., Jaggi, C. K., Bhunia, A. K., Shaikh, A. A., & Goh, M. (2017). Two-warehouse inventory model for non-instantaneous deteriorating items with stock-dependent demand and inflation using particle swarm optimization. Annals of Operations Research, 254, 401-423.

70.   Kumar, S., Chakraborty, D., & Malik, A. K. (2017). A two warehouse inventory model with stock-dependent demand and variable deterioration rate. International Journal of Future Revolution in Computer Science & Communication Engineering, 3(9), 20-24.

71.   Mishra, P., & Shaikh, A. (2017). Optimal ordering policy for an integrated inventory model with stock dependent demand and order linked trade credits for twin ware house system. Uncertain Supply Chain Management, 5(3), 169-186.

72.   Palanivel, M., Priyan, S., & Mala, P. (2018). Two-warehouse system for non-instantaneous deterioration products with promotional effort and inflation over a finite time horizon. Journal of Industrial Engineering International, 14, 603-612.

73.   Malik, A. K., Chakraborty, D., Bansal, K. K., & Kumar, S. (2017). Inventory model with quadratic demand under the two warehouse management system. International Journal of Engineering and Technology, 9(3), 2299-2303.

74.   Rastogi, M., Singh, S. R., Kushwah, P., & Tayal, S. (2017). Two warehouse inventory policy with price dependent demand and deterioration under partial backlogging. Decision science letters, 6(1), 11-22.

75.   Singh, S., Khurana, D., & Tayal, S. (2016). An economic order quantity model for deteriorating products having stock dependent demand with trade credit period and preservation technology. Uncertain Supply Chain Management, 4(1), 29-42.

76.   Singh, S. R., & Rathore, H. (2016). A two warehouse inventory model with preservation technology investment and partial backlogging. Scientia Iranica, 23(4), 1952-1958.

77.   Vikram, V., Ajay, T., Chandra, S., & Malik, A. K. (2016). A trade credit inventory model with multivariate demand for non-instantaneous decaying products. Indian Journal of Science and Technology, 9(15), 1-6.

78.   Vashisth, V., Soni, R., Jakhar, R., Sihag, D., & Malik, A. K. (2016, March). A two warehouse inventory model with quadratic decreasing demand and time dependent holding cost. In AIP Conference Proceedings (Vol. 1715, No. 1). AIP Publishing.

79.   Rashid, R., Hoseini, S. F., Gholamian, M. R., & Feizabadi, M. (2015). Application of queuing theory in production-inventory optimization. Journal of Industrial Engineering International, 11, 485-494.

80.   Chowdhury, R. R., Ghosh, S. K., & Chaudhuri, K. S. (2015). An inventory model for deteriorating items with stock and price sensitive demand. International Journal of Applied and Computational Mathematics, 1, 187-201.

81.   Palanivel, M., & Uthayakumar, R. (2016). Two-warehouse inventory model for non–instantaneous deteriorating items with optimal credit period and partial backlogging under inflation. Journal of Control and Decision, 3(2), 132-150.

82.   Mishra, S. S., Gupta, S., Yadav, S. K., & Rawat, S. (2015). Optimization of fuzzified Economic Order Quantity model allowing shortage and deterioration with full backlogging. American Journal of Operational Research, 5(5), 103-110.

83.   Jaggi, C. K., Tiwari, S., & Shafi, A. (2015). Effect of deterioration on two-warehouse inventory model with imperfect quality. Computers & Industrial Engineering, 88, 378-385.

84.   Bhunia, A. K., & Shaikh, A. A. (2015). An application of PSO in a two-warehouse inventory model for deteriorating item under permissible delay in payment with different inventory policies. Applied Mathematics and Computation, 256, 831-850.

85.   Das, D., Roy, A., & Kar, S. (2015). A multi-warehouse partial backlogging inventory model for deteriorating items under inflation when a delay in payment is permissible. Annals of Operations Research, 226, 133-162.

86.   Lee, C. C., & Ying, C. (2000). Optimal inventory policy for deteriorating items with two-warehouse and time-dependent demands. Production Planning & Control, 11(7), 689-696.

87.   Bhunia, A. K., Jaggi, C. K., Sharma, A., & Sharma, R. (2014). A two-warehouse inventory model for deteriorating items under permissible delay in payment with partial backlogging. Applied Mathematics and Computation, 232, 1125-1137.

88.   Agrawal, S., Banerjee, S., & Papachristos, S. (2013). Inventory model with deteriorating items, ramp-type demand and partially backlogged shortages for a two-warehouse system. Applied Mathematical Modelling, 37(20-21), 8912-8929.

89.   Singh, S.R., Gupta, V., & Goel, A. (2013)An EOQ Model with Preservation Technology Investment When Demand Depends on Selling Price and Credit Period Under Two Level of Trade Credit. Procedia Technology, 10, 227-235.

90.   Kumar, N., Singh, S. R., & Tomar, J. (2013). Two-warehouse inventory model with multivariate demand and K-release rule. Procedia Technology, 10, 788-796.

91.   Kumar, N., Singh, S. R., & Kumari, R. (2013). Two-warehouse inventory model of deteriorating items with three-component demand rate and time-proportional backlogging rate in fuzzy environment. International Journal of Industrial Engineering Computations, 4(4), 587-598.

92.   Gupta, K. K., Sharma, A., Singh, P. R., & Malik, A. K. (2013). Optimal ordering policy for stock-dependent demand inventory model with non-instantaneous deteriorating items. International Journal of Soft Computing and Engineering, 3(1), 279-281.

93.   Bhunia, A. K., Shaikh, A. A., & Gupta, R. K. (2015). A study on two-warehouse partially backlogged deteriorating inventory models under inflation via particle swarm optimisation. International Journal of Systems Science, 46(6), 1036-1050.

94.   Malik, A. K., & Singh, Y. (2013). A fuzzy mixture two warehouse inventory model with linear demand. International Journal of Application or Innovation in Engineering and Management, 2(2), 180-186.

95.   Sharma, A., Gupta, K. K., & Malik, A. K. (2013). Non-Instantaneous Deterioration Inventory Model with inflation and stock-dependent demand. International Journal of Computer Applications, 67(25).

96.   Niakan, F., Baboli, A., Botta-Genoulaz, V., Tavakkoli-Moghaddam, R., & Camapgne, J. P. (2013). A multi-objective mathematical model for green supply chain reorganization. IFAC Proceedings Volumes, 46(7), 81-86.

97.   Yang, H. L. (2012). Two-warehouse partial backlogging inventory models with three-parameter Weibull distribution deterioration under inflation. International Journal of Production Economics, 138(1), 107-116.

98.   Sett, B. K., Sarkar, B., & Goswami, A. (2012). A two-warehouse inventory model with increasing demand and time varying deterioration. Scientia Iranica, 19(6), 1969-1977.

99.   Dem, H., & Singh, S. R. (2012). A two-warehouse production model with quality consideration.Procedia Engineering, 38, 3242-3259.

100.      Yadav, D., Singh, S. R., & Kumari, R. (2012). Inventory model of deteriorating items with two-warehouse and stock dependent demand using genetic algorithm in fuzzy environment. YUJOR, 22(1), 51-78.

101.      Singh, S. R., Malik, A. K., & Gupta, S. K. (2011). Two warehouses inventory model with partial backordering and multi-variate demand under inflation. International Journal of Operations Research and Optimization, 2(2), 371-384.

102.      Singh, S. R., Kumari, R., & Kumar, N. (2011). A deterministic two warehouse inventory model for deteriorating items with stock-dependent demand and shortages under the conditions of permissible delay. International Journal of Mathematical Modelling and Numerical Optimisation, 2(4), 357-375.

103.      Singh, S. R., Malik, A. K., & Gupta, S. K. (2011). Two warehouses inventory model for non-instantaneous deteriorating items with stock-dependent demand. International Transactions in Applied Sciences, 3(4), 911-920.

104.      Liang, Y., & Zhou, F. (2011). A two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment. Applied Mathematical Modelling, 35(5), 2221-2231.

105.      Jaggi, C. K., Khanna, A., & Verma, P. (2011). Two-warehouse partial backlogging inventory model for deteriorating items with linear trend in demand under inflationary conditions. International Journal of Systems Science, 42(7), 1185-1196.

106.      Hariga, M. A. (2011). Inventory models for multi-warehouse systems under fixed and flexible space leasing contracts. Computers & Industrial Engineering, 61(3), 744-751.

107.      Malik, A. K., & Kumar, S. (2011). Two Warehouses Inventory Model with Multi-Variate Demand Replenishment Cycles and Inflation. International Journal of Physical Sciences, 23(3), 847-854.

108.      Singh, S. R., Malik, A. K., & Gupta, S. K. (2011). Two warehouses inventory model with partial backordering and multi-variate demand under inflation. International Journal of Operations Research and Optimization, 2(2), 371-384.

109.      Singh, S. R., & Malik, A. K. (2010). Inventory system for decaying items with variable holding cost and two shops. International Journal of Mathematical Sciences, 9(3-4), 489-511.

110.      Singh, S. R., Kumari, R., & Kumar, N. (2010). Replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial back logging with two-storage facility under inflation. International Journal of Operations Research and Optimization, 1(1), 161-179.

111.      Min, J., Zhou, Y. W., & Zhao, J. (2010). An inventory model for deteriorating items under stock-dependent demand and two-level trade credit. Applied Mathematical Modelling, 34(11), 3273-3285.

112.      Jaggi, C. K., & Verma, P. (2010). An optimal replenishment policy for non-instantaneous deteriorating items with two storage facilities. International Journal of Services Operations and Informatics, 5(3), 209-230.

113.      Uthayakumar, R., & Geetha, K. V. (2009). A Replenishment Policy for Non-instantaneous Deteriorating Inventory System with Partial Backlogging. Tamsui Oxford Journal of Mathematical Sciences (TOJMS), 25(3).

114.      Singh, S. R., & Malik, A. K. (2009). Two warehouses model with inflation induced demand under the credit period. International Journal of Applied Mathematical Analysis and Applications, 4(1), 59-70.

115.      Lee, C. C., & Hsu, S. L. (2009). A two-warehouse production model for deteriorating inventory items with time-dependent demands. European Journal of Operational Research, 194(3), 700-710.

116.      Geetha, K. V., & Uthayakumar, R. (2009). Optimal inventory control policy for items with time-dependent demand. American Journal of Mathematical and Management Sciences, 29(3-4), 457-476.

117.      Chung, K. J., Her, C. C., & Lin, S. D. (2009). A two-warehouse inventory model with imperfect quality production processes. Computers & Industrial Engineering, 56(1), 193-197.

118.      Singh, S. R., & Malik, A. K. (2009). Effect of inflation on two warehouse production inventory systems with exponential demand and variable deterioration. International Journal of Mathematical and Applications, 2(1), 141-149.

119.      Rong, M., Mahapatra, N. K., & Maiti, M. (2008). A two-warehouse inventory model for a deteriorating item with partially/fully backlogged shortage and fuzzy lead time. European Journal of Operational Research, 189(1), 59-75.

120.      Niu, B., & Xie, J. (2008). A note on “Two-warehouse inventory model with deterioration under FIFO dispatch policy”. European Journal of Operational Research, 190(2), 571-577.

121.      Jaggi, C. K., Aggarwal, K. K., & Verma, P. (2008). Two-Warehouse Inventory Model When Demand Is Price Sensitive. Indian Journal of'Mathematics and Mathematical Sciences, 53-65.

122.      Hsieh, T. P., Dye, C. Y., & Ouyang, L. Y. (2008). Determining optimal lot size for a two-warehouse system with deterioration and shortages using net present value. European Journal of Operational Research, 191(1), 182-192.

123.      Malik, A. K., Singh, S. R., & Gupta, C. B. (2008). An inventory model for deteriorating items under FIFO dispatching policy with two warehouse and time dependent demand. Ganita Sandesh, 22(1), 47-62.

124.      Roy, T., & Chaudhuri, K. S. (2007). An inventory model for a deteriorating item with price-dependent demand and special sale. International Journal of Operational Research, 2(2), 173-187.

125.      Dye, C. Y., Ouyang, L. Y., & Hsieh, T. P. (2007). Deterministic inventory model for deteriorating items with capacity constraint and time-proportional backlogging rate. European Journal of Operational Research, 178(3), 789-807.

126.      Huq, F., Cutright, K., Jones, V., & Hensler, D. A. (2006). Simulation study of a two‐level warehouse inventory replenishment system. International Journal of Physical Distribution & Logistics Management, 36(1), 51-65.

127.      Ghosh, S. K., & Chaudhuri, K. S. (2006). An EOQ model with a quadratic demand, time-proportional deterioration and shortages in all cycles. International Journal of Systems Science, 37(10), 663-672.

128.      Zhou, Y. W., & Yang, S. L. (2005). A two-warehouse inventory model for items with stock-level-dependent demand rate. International Journal of Production Economics, 95(2), 215-228.

129.      Lee, C. C. (2006). Two-warehouse inventory model with deterioration under FIFO dispatching policy. European Journal of Operational Research, 174(2), 861-873.

130.      Yang, H. L. (2004). Two-warehouse inventory models for deteriorating items with shortages under inflation. European Journal of Operational Research, 157(2), 344-356.

131.      Zhou, Y. W. (2003). A multi-warehouse inventory model for items with time-varying demand and shortages. Computers & Operations Research, 30(14), 2115-2134.

132.      Khanra, S., & Chaudhuri, K. S. (2003). A note on an order-level inventory model for a deteriorating item with time-dependent quadratic demand. Computers & Operations Research, 30(12), 1901-1916.

133.      Kar, S., Bhunia, A. K., & Maiti, M. (2001). Deterministic inventory model with two levels of storage, a linear trend in demand and a fixed time horizon. Computers & Operations Research, 28(13), 1315-1331.

134.      Lee, C. C., & Ying, C. (2000). Optimal inventory policy for deteriorating items with two-warehouse and time-dependent demands. Production Planning & Control, 11(7), 689-696.

135.      Teng, J. T., Chern, M. S., Yang, H. L., & Wang, Y. J. (1999). Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand. Operations Research Letters, 24(1-2), 65-72.

136.      Bhunia, A. K., & Maiti, M. (1998). A two warehouse inventory model for deteriorating items with a linear trend in demand and shortages. Journal of the Operational Research Society, 49(3), 287-292.

137.      Giri, B. C., & Chaudhuri, K. S. (1997). Heuristic models for deteriorating items with shortages and time-varying demand and costs. International journal of systems science, 28(2), 153-159.

138.      Sarker, B. R., Mukherjee, S., & Balan, C. V. (1997). An order-level lot size inventory model with inventory-level dependent demand and deterioration. International Journal of Production Economics, 48(3), 227-236.

139.      Benkherouf, L. (1997). A deterministic order level inventory model for deteriorating items with two storage facilities. International journal of production economics, 48(2), 167-175.

140.      Chen, F., & Zheng, Y. S. (1997). One-warehouse multiretailer systems with centralized stock information. Operations Research, 45(2), 275-287.

141.      Hariga, M. (1996). Optimal EOQ models for deteriorating items with time-varying demand. Journal of the Operational Research Society, 47(10), 1228-1246.

142.      Ishii, H., & Nose, T. (1996). Perishable inventory control with two types of customers and different selling prices under the warehouse capacity constraint. International Journal of Production Economics, 44(1-2), 167-176.

143.      Padmanabhan, G., & Vrat, P. (1995). EOQ models for perishable items under stock dependent selling rate. European Journal of Operational Research, 86(2), 281-292.

144.      Ahire, S. L., & Schmidt, C. P. (1996). A model for a mixed continuous-periodic review one-warehouse, N-retailer inventory system. European Journal of Operational Research, 92(1), 69-82.

145.      Hariga, M. A., & Benkherouf, L. (1994). Optimal and heuristic inventory replenishment models for deteriorating items with exponential time-varying demand. European Journal of Operational Research, 79(1), 123-137.

146.      Pakkala, T. P. M., & Achary, K. K. (1992). A deterministic inventory model for deteriorating items with two warehouses and finite replenishment rate. European Journal of Operational Research, 57(1), 71-76.

147.      Pakkala, T. P. M., & Achary, K. K. (1991). A two-warehouse probabilistic order-level inventory model for deteriorating items. Journal of the Operational Research Society, 42(12), 1117-1122.

148.      Padmanabhan, G. A., & Vrat, P. (1990). An EOQ model for items with stock dependent consumption rate and exponential decay. Engineering Costs and Production Economics, 18(3), 241-246.

149.      Sarma, K. V. S. (1987). A deterministic order level inventory model for deteriorating items with two storage facilities. European journal of operational research, 29(1), 70-73.

150.      Schwarz, L. B., Deuermeyer, B. L., & Badinelli, R. D. (1985). Fill-rate optimization in a one-warehouse N-identical retailer distribution system. Management science, 31(4), 488-498.

151.      Goyal, S. K. (1985). Economic order quantity under conditions of permissible delay in payments. Journal of the operational research society, 335-338.